It's always good to show a photo that includes the problem area plus some surroundings for context. We may see something relevant that you don't mention.

By "french drain," I presume you mean a drain that empties out at some lower elevation ...?

A "dry well" is likely to fill up in a heavy rain.

In clay soil, a dry well isn't particularly useful. You're just taking an above grade puddle and moving part of it underground. As Yard mentioned, the french drain could be effective if it daylights somewhere. If all it is is a long piece of perforated pipe in a trench with no way for water to exit, nothing is solved.

Depending on where you're located, your local county government may have info on current stormwater best management practices (BMP) online. Rain gardens, infiltration trenches, and other means of moving and dispersing water can be effective if designed and installed by someone knowledgeable (but they can be swampy disasters if done by a random Chuck in a truck)

Thank you both for the input! That was essentially what I had found as well. The plan for the french drain would be to have it run all the way out to the drainage ditch by the street. Here was the proposed plan, as well as some pictures of the area.

Water cannot escape from this corner of the yard by running in some frontward oriented direction? If "no," and it were my problem, I would not like the idea of running an underground drain line the full length of the property. I would be considering raising the corner of the yard enough that it could drain via surface route. It looks like all the paving is on sand so wouldn't be that difficult to pull up and reset. It might even be looked at as an opportunity for creating a better patio shape. I would examine raising the barbecue grill pieces higher by building on top of what's there now with a little remodeling. A dry well would not likely solve the problem long term.

Insofar as your landscape plan goes, as mentioned before I'd avoid the idea of near complete enclosure by plants. I would also clean it up by eliminating loopiness of the bed line.